International Journal of Analysis and Applications Volume 18, Number 6 (2020), 998-1014 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-998 FUZZINESS AND ROUGHNESS IN HYPERQUANTALES RAEES KHAN1,∗, MASLINA DARUS2,∗, MUHAMMAD FAROOQ3, ASGHAR KHAN3 AND NASIR KHAN1 1Department of Mathematics, FATA University, TSD Darra NMD Kohat, KP, Pakistan 2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia 3Department of Mathematics, Abdul Wali Khan University, Mardan, KP, Pakistan ∗Corresponding authors: raeeskhan@fu.edu.pk, maslina@ukm.edu.my Abstract. Theories of fuzzy set and rough set are powerful mathematical tools for modelling various types of uncertainty. In this paper, we introduce the notions of bi-hyperideal, fuzzy bi-hyperideals of hyperquantales and their related properties is given. Furthermore we introduce the notion of generalized rough fuzzy bi-hyperideals. Moreover, we will describe the set-valued homomorphism and strong set-valued homomorphism of hyperquantales and some related properties will be study. 1. Introduction The theory of rough sets was introduced by Pawlak [15, 16], to deal with uncertain knowledge in information systems. The rough set theory has been emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy or incomplete information. It has turned out to be fundamentally important in artificial intelligence and cognitive sciences, especially in fields such as machine learning, knowl- edge acquisition, decision analysis, expert systems, pattern recognition. With the development of rough set theory, possible connections between rough sets and various algebraic systems were considered by many au- thors. Inspired by the construction of Pawlak rough set algebras and the investigation in algebraic properties Received July 7th, 2020; accepted August 24th, 2020; published October 7th, 2020. 2010 Mathematics Subject Classification. 03E72. Key words and phrases. rough set; fuzzy set; hyperquantale; fuzzy bi-hyperideal; generalized rough fuzzy bi-hyperideal. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 998 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-998 Int. J. Anal. Appl. 18 (6) (2020) 999 of rough sets in [17,18]. As a combination of algebraic structures and partially ordered structures, the theory of quantales was initiated by Mulvey [20] to study the spectrum of C-algebras and the foundations of quan- tum mechanics. Wang and Zhao [22, 23] proposed the concepts of ideals and prime ideals of quantales. Yang and Xu [24], considered quantales as universal sets and introduced the notions of rough (prime, semi-prime, primary) ideals and prime radicals of upper rough ideals of quantales. Wang in [21] studied prime radical theorem in quantales. The concept of fuzzy sets was introduced by Zadeh [19] in 1965. The theory of fuzzy sets has been developed fast and has many applications in many branches of sciences. Luo and Wang in [25], studied roughness and fuzziness in quantales. Davvaz et al. in [6] applied Atanassov’s intuitionistic fuzzy set theory to quantales. In [29, 30], Saqib and Shabir studied relationship between generalized rough sets and quantale by using fuzzy ideals of quantale. Algebraic hyperstructures represent a natural extension of classical algebraic structures and they were orig- inally proposed in 1934 by a French mathematician Marty [1], at the 8th Congress of Scandinavian Mathe- maticians. One of the main reason which attracts researches towards hyperstructures is its unique property that in hyperstructures composition of two elements is a set, while in classical algebraic structures the com- position of two elements is an element. Thus algebraic hyperstructures are natural extension of classical algebraic structures. Since then, hyperstructures are widely investigated from the theoretical point of view and for their applications to many branches of pure and applied mathematics (see [2–5, 8]). Since then, there appeared many components of hyperalgebras such as hypergroups in [9], hyperrings etc in [10, 11]. Konstantinidou and Mittas have introduced the concept of hyperlattices in [12], fuzzy ideal of hyperlattices have been introduced in [7]. The notion of hyperlattices is a generalization of the notion of lattices and there are some intimate connections between hyperlattices and lattices. In particular, Rasouli and Davvaz further studied the theory of hyperlattices and obtained some interesting results [13, 14], which enrich the theory of hyperlattices. In [28], Estaji and Bayati studied rough Sets in terms of Hyperlattices. In [26], Khan et al. introduced the notions of hyperideals and fuzzy hyperideals of hyperquantales. In this paper, we introduce the notions of bi-hyperideal and fuzzy bi-hyperideals of hyperquantales and give several characterizations. In addition, we will introduce the notions of generalized rough fuzzy bi-hyperideal in hyperquantales and some new properties will be obtain. 2. Preliminaries A map ∗ : S ×S → P∗(S) is called hyperoperation or join operation on the set S, where S is a non-empty set and P∗(S) = P(S)\{∅} denotes the set of all non-empty subsets of S. A hyperstructure is called the pair (S,∗) where ∗ is a hyperoperation on the set S. Int. J. Anal. Appl. 18 (6) (2020) 1000 Definition 2.1. (see [26]). A hyperquantale is a complete hyperlattice Q with an associative binary operation ∗ satisfying x∗ (∨ i∈I yi ) = ∨ i∈I (x∗yi) , (∨ i∈I xi ) ∗ y = ∨ i∈I (xi ∗y) for all x,y,xi,yi ∈ Q (i ∈ I) where I is an index set. A hyperquantale Q is called commutative if x∗y = y ∗x for all x,y ∈ Q. Throughout this paper, we denote the least and greatest elements of a hyperquantale denoted by ⊥ and > respectively. Definition 2.2. (see [26]). Let Q be a hyperquantale. A non-empty subset A of Q is called a left (resp. right) hyperideal of Q if it satisfies the following conditions: (1) x,y ∈ A implies x∨y ⊆ A. (2) (∀ x,y ∈ Q) x ∈ A and y ≤ x imply y ∈ A. (3) ∀ x ∈ Q and a ∈ A, we have x∗a ⊆ A (resp. a∗x ⊆ A). A non empty subset A of Q is called a two sided hyperideal or simply a hyperideal of Q if it is both a left hyperideal and right hyperideal of Q. Definition 2.3. Let Q be a hyperquantale. A non-empty subset B of Q is called a bi-hyperideal of Q if it satisfies the following conditions: (1) x,y ∈ B implies x∨y ⊆ B. (2) x,y ∈ B implies x∗y ⊆ B. (3) (∀ x,y ∈ Q) x ∈ B and y ≤ x imply y ∈ B. (4) ∀ y ∈ Q and x,z ∈ B, we have x∗y ∗z ⊆ B. Example 2.1. Let Q = {⊥,e1,e2,e3,>} and define ∗ and ∨ by the following Cayley tables: ∗ ⊥ e1 e2 e3 > ⊥ {⊥} {⊥} {⊥} {⊥} {⊥} e1 {⊥} {e1} {e1} {e1} {e1} e2 {⊥} {e1} {e2} {e1} {e2} e3 {⊥} {e1} {e1} {e3} {e3} > {⊥} {e1} {e2} {e3} {>} Int. J. Anal. Appl. 18 (6) (2020) 1001 and ∨ ⊥ e1 e2 e3 > ⊥ {⊥} {e1} {e2} {e3} {>} e1 {e1} {⊥,e1} {⊥,e2} {⊥,e3} {⊥,>} e2 {e2} {⊥,e2} {⊥,e1,e2} {⊥,>} {⊥,e3,>} e3 {e3} {⊥,e3} {⊥,>} {⊥,e1,e3} {⊥,e2,>} > {>} {⊥,>} {⊥,e3,>} {⊥,e2,>} {⊥,e1,e2,e3,>} Thus all bi-hyperideals of Q are {⊥} , {⊥,e1} , {⊥,e1,e2} , {⊥,e1,e3} and Q. For A,B ⊆ Q, we have A∗B := ⋃ {a∗ b : a ∈ A, b ∈ B} and A∨B := ⋃ {a∨ b : a ∈ A, b ∈ B}. For A ⊆ Q,we denote (A] := {a ∈ Q : a ≤ b for some b ∈ A}. 3. Fuzzy hyperideals of hyperquantale Let Q be a hyperquantale. A function f from a nonempty set X to the unit interval [0, 1] is called a fuzzy subset of Q. Let Q be a hyperquantale and f be a fuzzy subset of Q. Then for every t ∈ [0, 1] the set U (f; t) = {x | x ∈ Q, f (x) ≥ t} . For x ∈ Q, we define Ax = {(y,z) ∈ Q×Q | x ≤ y ∗z}. Definition 3.1. (see [26]). Let Q be a hyperquantale and f,g are any two fuzzy subsets of Q. We define the product f ◦g of f and g as follows: (f ◦g) (x) =   ∨ (y,z)∈Ax {f (y) ∧ g (z)} , if Ax 6= ∅ 0, if Ax = ∅ . For two functions f and g then f ⊆ g if and only if f (x) ≤ g (x) . Let Q be a hyperquantale and ∅ 6= A ⊆ Q. Then the characteristic function χA of A is defined as: χA : Q −→ [0, 1] ,−→ χA (x) =   1 if x ∈ A0 if x /∈ A Definition 3.2. (see [26]). Let Q be a hyperquantale. A fuzzy subset f of Q is called a fuzzy subhyperquantale of Q if it satisfies the following conditions: (1) (∀x,y ∈ Q) ∧ α∈x∗y f (α) ≥ f (x) ∧ f (y) . (2) (∀x,y ∈ Q) ∧ β∈x∨y f (β) ≥ f (x) ∧ f (y) . Int. J. Anal. Appl. 18 (6) (2020) 1002 Definition 3.3. (see [26]). Let Q be a hyperquantale. A fuzzy subset f of Q is called a fuzzy left (resp. right) hyperideal of Q if it satisfies the following conditions: (1) (∀x,y ∈ Q) ∧ α∈x∗y f (α) ≥ f (y) (resp. ∧ α∈x∗y f (α) ≥ f (x)). (2) (∀x,y ∈ Q) ∧ β∈x∨y f (β) ≥ f (x) ∧ f (y) . (3) (∀x,y ∈ Q) x ≤ y then f (x) ≥ f (y) . 4. Fuzzy bi-hyperideal of hyperquantale In this section, we introduce the notion of fuzzy bi-hyperideal of hyperquantale and investigate some related properties. Definition 4.1. Let Q be a hyperquantale. A fuzzy subset f of Q is called a fuzzy bi-hyperideal of Q if it satisfies the following conditions: (1) (∀x,y ∈ Q) ∧ α∈x∗y f (α) ≥ f (x) ∧ f (y) . (2) (∀x,y ∈ Q) ∧ β∈x∨y f (β) ≥ f (x) ∧ f (y) . (3) (∀x,y,z ∈ Q) ∧ γ∈(x∗y∗z) f (γ) ≥ f (x) ∧ f (z) . (4) (∀x,y ∈ Q) x ≤ y then f (x) ≥ f (y) . Example 4.1. Let Q = {⊥,e1,e2,>} and define ∗ and ∨ by the following Cayley tables: ∗ ⊥ e1 e2 > ⊥ {⊥} {⊥} {⊥} {⊥} e1 {⊥} {e1} {⊥} {e1} e2 {⊥} {⊥} {e2} {e2} > {⊥} {e1} {e2} {>} and ∨ ⊥ e1 e2 > ⊥ {⊥} {e1} {e2} {>} e1 {e1} {⊥,e1} {>} {e2,>} e2 {e2} {>} {⊥,e2} {e1,>} > {>} {e2,>} {e1,>} Q Let us define a fuzzy subset f : Q −→ [0, 1] as follows: f (x) =   1 if x = ⊥0.4 if x ∈{e1,e2,>} Then it is easy to verify that f is a fuzzy bi-hyperideal of Q. Int. J. Anal. Appl. 18 (6) (2020) 1003 Theorem 4.1. Let B be a non empty subset of a hyperquantale Q. Then B is a bi-hyperideal of Q if and only if χB is a fuzzy hyperideal of Q. Proof. Suppose that B is a hyperideal of Q. Let x,y ∈ Q. If x,y ∈ B then x∨y ⊆ B and x∗y ⊆ B. Since x,y ∈ B, we have χB (x) = χB (y) = 1, for any α ∈ x∨y ⊆ B, we have ∧ α∈x∨y χB (α) = 1 = χB (x) ∧ χB (y) . Also for any β ∈ x∗y ⊆ B, we have ∧ β∈x∗y χB (β) = 1 = χB (x) ∧ χB (y) . If x /∈ B or y /∈ B. Then x∨y ⊆ B or x∨y * B and x∗y ⊆ B or x∗y * B. In all the cases we have χB (x) ∧ χB (y) = 0 ≤ ∧ α∈x∨y χB (α) and χB (x) ∧ χB (y) = 0 ≤ ∧ β∈x∗y χB (β) . Let now x,y ∈ Q, x ≤ y. Then χB (x) ≥ χB(y). In fact, if y ∈ B, then χB(y) = 1. Since Q 3 x ≤ y ∈ B, by hypothesis we have x ∈ B, then χB (x) = 1. Thus χB (x) ≥ χB (y) . If y /∈ B, then χB(y) = 0. Since x ∈ Q, we have χB (x) ≥ 0 = χB(y). Let x,y and z be any elements of Q. If x,z ∈ B, then χB (x) = χB (z) = 1 and since for every α ∈ x ∗ y ∗ z ⊆ B, we have χB (α) = 1 = χB (x) ∧ χB (z) . Thus ∧ α∈x∗y∗z χB (α) = 1 = χB (x) ∧ χB (z) . If x /∈ B or z /∈ B, then χB (x) = 0 or χB (z) = 0, and so we have χB (α) ≥ 0 = χB (x) ∧ χB (z) . Thus ∧ α∈x∗y∗z χB (α) ≥ χB (x) ∧ χB (z) . Therefore χB is a fuzzy bi-hyperideal of Q. Conversely, assume that χB is a fuzzy bi-hyperideal of Q. Let x,y ∈ B. Then ∧ α∈x∨y χB (α) = χB (x) ∧ χB (y) = 1, and thus α ∈ x∨y ⊆ B. Since x,y ∈ B. Then for any z ∈ x∗y, we have ∧ z∈x∗y χA (z) ≥ χA (x) ∧ χA (y) = 1. Implies that ∧ z∈x∗y χA (z) = 1. Thus x∗y ⊆ A. If x ≤ y and y ∈ B, then χB (x) ≥ χB (y) = 1, implies that x ∈ B. Let x,z ∈ B and y ∈ S such that for any α ∈ x∗y ∗z, we have Since∧ α∈x∗y∗z χB (α) ≥ χB (x) ∧ χB (z) = 1 ∧ 1 = 1. Hence for each α ∈ x ∗ y ∗ z, we have χB (α) = 1, and so α ∈ B. Thus x ∗ y ∗ z ⊆ B. Thus B is a bi-hyperideal of Q. � Theorem 4.2. Let Q be a hyperquantale. A fuzzy subset f of Q is a fuzzy bi-hyperideal of Q if and only if for each t ∈ [0, 1], U(f; t) 6= ∅ is a bi-hyperideal of Q. Proof. Assume that U(f; t) is a bi-hyperideal of Q. Let x,y ∈ Q such that x ≤ y. If f (y) = 0 then f (x) ≥ f (y) . If f (y) = t then y ∈ U(f; t). Since x ≤ y and U(f; t) is a bi-hyperideal of Q, we have x ∈ U(f; t). Then f (x) ≥ t = f (y) . Since U(f; t) 6= ∅ is a bi-hyperideal of Q. If ∧ α∈x∗y f (α) < f (x) ∧ f (y) for some x,y ∈ Q, then there exists t0 ∈ [0, 1] such that ∧ α∈x∗y f (α) < t0 ≤ f (x) ∧ f (y) , which implies that x,y ∈ U(f; t) and x∗y * U(f; t). It contradicts the fact that U(f; t) is a bi-hyperideal of Q. Consequently,∧ α∈x∗y f (α) ≥ f (x) ∧ f (y) for all x,y ∈ Q. Next we show that ∧ B∈x∨y f (α) ≥ f (x) ∧ f (y) for all x,y ∈ Q. If there exist x,y ∈ Q and t0 ∈ [0, 1] such that ∧ β∈x∨y f (β) < t0 ≤ f (x) ∧ f (y). Then x,y ∈ U(f; t) and Int. J. Anal. Appl. 18 (6) (2020) 1004 β ∈ x∨y * U(f; t). It is again contradicts the fact that U(f; t) is a bi-hyperideal of Q. Thus ∧ β∈x∨y f (β) ≥ f (x) ∧ f (y) . Now let x,y,z ∈ U(f; t). Then x ∗ y ∗ z ⊆ U(f; t). Since x,z ∈ U(f; t). Then f (x) ≥ t and f (z) ≥ t. So for any α ∈ x∗y ∗z, we have f (α) ≥ t. Thus f (x) ∧ f (z) = t ≤ ∧ α∈x∗y∗z f (α). Therefore f is a fuzzy bi-hyperideal of Q. Conversely, suppose that f be a fuzzy bi-hyperideal of Q. Let x,y ∈ U(f; t). Then f(x) ≥ t, f(y) ≥ t. Since f is a fuzzy bi-hyperideal of Q, so we have ∧ α∈x∗y f (α) ≥ f(x) ∧ f (y) = t. Hence f(α) ≥ t for all α ∈ x ∗ y, this implies α ∈ U(f; t) that is x ∗ y ⊆ U(f; t). As f is a fuzzy bi-hyperideal of Q. Then∧ w∈x∨y f (w) ≥ f (x) ∧ f (y) ≥ t. Hence f (w) ≥ t for any w ∈ x ∨ y implies that w ∈ U(f; t). Thus x ∨ y ⊆ U(f; t). Now let x,y,z ∈ U(f; t). Then f(x) ≥ t, f(y) ≥ t and f(z) ≥ t. Since f is a fuzzy bi-hyperideal of Q, we have ∧ β∈x∗y∗z f (β) ≥ f (x) ∧ f (z) = t. So f (β) ≥ t. Hence x ∗ y ∗ z ⊆ U(f; t). Let x ∈ U(f; t) and y ∈ Q with y ≤ x. Then t ≤ f (x) ≤ fA (y) , we get y ∈ U(f; t). Therefore U(f; t) is a bi-hyperideal of Q. � Theorem 4.3. Let {fi | i ∈ I} be a family of fuzzy bi-hyperideals of Q. Then f = ⋂ i∈I fi is a fuzzy bi- hyperideal of Q where (⋂ i∈I fi ) (x) = ∧ i∈I (fi (x)) . Proof. Let x,y ∈ Q. Then, since each fi (i ∈ I) is a fuzzy bi-hyperideal of Q, so ∧ α∈x∨y fi (α) ≥ fi (x) ∧ fi (y) . Thus for any α ∈ x∨y, fi (α) ≥ fi (x) ∧ fi (y) , and we have f (α) = (⋂ i∈I fi ) (α) = ∧ i∈I (fi (α)) ≥ ∧ i∈I ( fi (x) ∧ fi (y) ) = (∧ i∈I (fi (x)) )∧(∧ i∈I (fi (y)) ) = (⋂ i∈I fi ) (x) ∧(⋂ i∈I fi ) (y) = f (x) ∧ f (y) , which implies that ∧ α∈x∨y f (α) ≥ f (x) ∧ f (y) . Let β ∈ x∗y and ∧ β∈x∗y fi (β) ≥ fi (x) ∧ fi (y) . Thus for any β ∈ x∗y, fi (β) ≥ fi (x) ∧ fi (y) . Then Int. J. Anal. Appl. 18 (6) (2020) 1005 f (β) = (⋂ i∈I fi ) (β) = ∧ i∈I (fi (β)) ≥ ∧ i∈I ( fi (x) ∧ fi (y) ) = (⋂ i∈I fi ) (x) ∧(⋂ i∈I fi ) (y) = f (x) ∧ f (y) . Thus ∧ β∈x∗y f (β) ≥ f (x) ∧ f (y). Now let x,y,z ∈ Q. Then for any γ ∈ x∗y ∗z, we have f (γ) = (⋂ i∈I fi ) (γ) = ∧ i∈I (fi (γ)) ≥ ∧ i∈I ( fi (x) ∧ f (z) ) = (∧ i∈I fi (x) )∧(∧ i∈I fi (z) ) = (⋂ i∈I fi ) (x) ∧(⋂ i∈I fi ) (z) = f (x) ∧ f (z) . Thus ∧ γ∈x∗y∗z f (γ) ≥ f (x) ∧ f (z) . Furthermore, if x ≤ y, then f (x) ≥ f (y) . Indeed: Since every fi (i ∈ I) is a fuzzy bi-hyperideal of Q, it can be obtained that fi (x) ≥ fi (y) for all i ∈ I. Thus f (x) = (⋂ i∈I fi ) (x) = ∧ i∈I (fi (x)) ≥ ∧ i∈I (fi (y)) = (⋂ i∈I fi ) (y) = f (y) . Thus f = ⋂ i∈I fi is a fuzzy bi-hyperideal of Q. � Int. J. Anal. Appl. 18 (6) (2020) 1006 5. Homomorphism and generalized rough fuzzy bi-hyperideals of hyperquantales Definition 5.1. (see [27]). Let X and Y be two nonempty universes. Let F be a set-valued mapping given by F : X −→P (Y ), where P (Y ) is the power set of Y . Then the triple (X,Y,F) is referred to as a generalized approximation space or generalized rough set. Any set-valued function from X to P (Y ) defines a binary relation from X to Y by setting ρF = {(a,b) | b ∈ F (a)}. Obviously, if ρ is an arbitrary relation from X to Y , then a set-valued mapping Fρ : X −→P (Y ) can be defined by Fρ (a) = {b ∈ Y | (a,b) ∈ ρ} where a ∈ X. For any set A ⊆ Y , the lower and upper approximations represented by F− (A) and F+ (A) respectively, are defined as F− (A) = {a ∈ X | F (a) ⊆ A} , F+ (A) = {a ∈ X | F (a) ∩A 6= ∅} . We call the pair (F− (A) ,F+ (A)) generalized rough set, and F−,F+ are termed as lower and upper generalized approximation operators, respectively. Definition 5.2. Let (Q1,∗1) and (Q2,∗2) be two hyperquantales. A set-valued mapping F : Q1 −→P∗ (Q2) , where P∗ (Q2) represents the collection of all nonempty subsets of Q2 is called a set-valued homomorphism if, for all ai,a,b ∈ Q1 (i ∈ I) , (1) F (a) ∗2 F (b) ⊆ F (a∗1 b) . (2) ∨ i∈I F (ai) ⊆ F (∨ i∈I ai ) . A set-valued mapping F : Q1 −→P∗ (Q2) is called a strong set-valued homomorphism if we replace ⊆ by = in (1) and (2). Definition 5.3. Let (Q1,∗1) and (Q2,∗2) be two hyperquantales and let F be a set-valued homomorphism. Let f be any fuzzy subset of Q2. Then for every x ∈ Q1, we defines F− (f) (x) = ∧ y∈F(x) f (y) , F+ (f) (x) = ∨ y∈F(x) f (y) . Here F− (f) is the generalized lower approximation and F+ (f) is the generalized upper approximation of the fuzzy subset of f. The pair (F− (f) ,F+ (f)) is called generalized rough fuzzy subset of Q1, if F − (f) 6= F+ (f) . Definition 5.4. Let F be a set-valued homomorphism. A fuzzy subset f of the hyperquantale Q2 is called a lower (resp. upper) generalized rough fuzzy bi-hyperideal of Q2 if F − (f) (resp. F+ (f)) is a fuzzy bi- hyperideal of Q1. A fuzzy subset f of Q2, which is both an upper and a lower generalized rough fuzzy bi-hyperideal of Q2, is called generalized rough fuzzy bi-hyperideal of Q2. Int. J. Anal. Appl. 18 (6) (2020) 1007 Theorem 5.1. Let F be a strong set-valued homomorphism and let f be a fuzzy bi-hyperideal of Q2. Then set F− (f) is a fuzzy bi-hyperideal of Q1. Proof. Assume that f is a fuzzy bi-hyperideal of Q2, then we have ∧ α∈x∨y f (α) ≥ f (x) ∧ f (y) imply that f (α) ≥ f (x) ∧ f (y) ∀x,y ∈ Q2 and α ∈ x∨y. Also F is a strong set-valued homomorphism, so F (x∨y) = F (x) ∨F (y) ∀x,y ∈ Q1. Therefore for any α ∈ x∨y F− (f) (α) = F− (f) (x∨y) = ∧ α∈F(x∨y) f (α) = ∧ α∈F(x)∨F(y) f (α) . Since α ∈ F (x) ∨F (y), there exist a ∈ F (x) and b ∈ F (y) such that α ∈ a∨ b. Hence F− (f) (x∨y) = ∧ a∨b∈F(x)∨F(y) f (a∨ b) ≥ ∧ a∈F(x),b∈F(y) ( f (a) ∧ f (b) ) =     ∧ a∈F(x) f (a)  ∧   ∧ b∈F(y) f (b)     = F− (f) (x) ∧ F− (f) (y) . Hence ∧ α∈x∨y F− (f) (α) ≥ F− (f) (x) ∧ F− (f) (y) ∀x,y ∈ Q1. Again since F is a strong set-valued homomorphism, so we have F (x∗1 y) = F (x) ∗2 F (y) ∀x,y ∈ Q1. Thus for any β ∈ x∗1 y we have, F− (f) (x∗1 y) = ∧ β∈F(x∗1y) f (β) = ∧ β∈F(x)∗2F(y) f (β) . Since β ∈ F (x) ∗2 F (y), there exist a ∈ F (x) and b ∈ F (y) such that β ∈ a∗2 b. Hence F− (f) (β) = F− (f) (x∗1 y) = ∧ a∗2b∈F(x)∗2F(y) f (a∗2 b) ≥ ∧ a∈F(x),b∈F(y) ( f (a) ∧ f (b) ) =     ∧ a∈F(x) f (a)  ∧   ∧ b∈F(y) f (b)     = F− (f) (x) ∧ F− (f) (y) . Hence ∧ β∈x∗1y F− (f) (β) ≥ F− (f) (x) ∧ F− (f) (y) ∀x,y ∈ Q1. Again since f is a fuzzy bi-hyperideal of Q2, so for any γ ∈ x∗1 y ∗1 z, we have Int. J. Anal. Appl. 18 (6) (2020) 1008 F− (f) (γ) = F− (f) (x∗1 y ∗1 z) = ∧ γ∈F(x∗1y∗1z) f (γ) = ∧ γ∈(F(x)∗2F(y)∗2F(z)) f (γ) . Since γ ∈ F (x)∗2 F (y)∗2 F (z), there exist a ∈ F (x) and b ∈ F (y) and c ∈ F (z) such that γ ∈ a∗2 b∗2 c. Hence F− (f) (γ) = F− (f) (x∗1 y ∗1 z) = ∧ a∗2b∗2c∈F(x)∗2F(y)∗2F(z) f (a∗2 b∗2 c) ≥ ∧ a∈F(x),c∈F(z) ( f (a) ∧ f (c) ) =     ∧ a∈F(x) f (a)  ∧   ∧ c∈F(z) f (c)     = F− (f) (x) ∧ F− (f) (z) . Hence ∧ γ∈(x∗1y∗1z) F− (f) (γ) ≥ F− (f) (x) ∧ F− (f) (z) ∀x,y,z ∈ Q1. � Theorem 5.2. Let F be a strong set-valued homomorphism and let f be a fuzzy bi-hyperideal of Q2. Then F+ (f) is a fuzzy bi-hyperideal of Q1. Proof. Assume that f is a fuzzy bi-hyperideal of Q2, then we have ∧ α∈x∨y f (α) ≥ f (x) ∧ f (y) imply that f (α) ≥ f (x) ∧ f (y) ∀x,y ∈ Q2 and α ∈ x∨y. Also F is a strong set-valued homomorphism, so F (x∨y) = F (x) ∨F (y) ∀x,y ∈ Q1. Therefore for any α ∈ x∨y F+ (f) (α) = F+ (f) (x∨y) = ∨ α∈F(x∨y) f (α) = ∨ α∈F(x)∨F(y) f (α) . Since α ∈ F (x) ∨F (y), there exist a ∈ F (x) and b ∈ F (y) such that α ∈ a∨ b. Hence F+ (f) (α) = F+ (f) (x∨y) = ∨ a∨b∈F(x)∨F(y) f (a∨ b) ≥ ∨ a∈F(x),b∈F(y) ( f (a) ∧ f (b) ) =     ∨ a∈F(x) f (a)  ∧   ∨ b∈F(y) f (b)     = F+ (f) (x) ∧ F+ (f) (y) . Hence ∧ α∈x∨y F+ (f) (α) ≥ F+ (f) (x) ∧ F+ (f) (y) ∀x,y ∈ Q1. Again since F is a strong set-valued homomorphism, so we have F (x∗1 y) = F (x) ∗2 F (y) ∀x,y ∈ Q1. Thus for any β ∈ x∗1 y we have, Int. J. Anal. Appl. 18 (6) (2020) 1009 F+ (f) (β) = F+ (f) (x∗1 y) = ∨ β∈F(x∗1y) f (β) = ∨ β∈F(x)∗2F(y) f (β) . Since β ∈ F (x) ∗2 F (y), there exist a ∈ F (x) and b ∈ F (y) such that β ∈ a∗2 b. Hence F+ (f) (β) = F+ (f) (x∗1 y) = ∨ a∗2b∈F(x)∗2F(y) f (a∗2 b) ≥ ∨ a∈F(x),b∈F(y) ( f (a) ∧ f (b) ) =     ∨ a∈F(x) f (a)  ∧   ∨ b∈F(y) f (b)     = F+ (f) (x) ∧ F+ (f) (y) . Hence ∧ β∈x∗1y F+ (f) (β) ≥ F+ (f) (x) ∧ F+ (f) (y) ∀x,y ∈ Q1. Again since f is a fuzzy bi-hyperideal of Q2, so for any γ ∈ x∗1 y ∗1 z, we have F+ (f) (γ) = F+ (f) (x∗1 y ∗1 z) = ∨ γ∈F(x∗1y∗1z) f (γ) = ∨ γ∈(F(x)∗2F(y)∗2F(z)) f (γ) . Since γ ∈ F (x)∗2 F (y)∗2 F (z), there exist a ∈ F (x) and b ∈ F (y) and c ∈ F (z) such that γ ∈ a∗2 b∗2 c. Hence F+ (f) (γ) = F+ (f) (x∗1 y ∗1 z) = ∨ a∗2b∗2c∈(F(x)∗2F(y)∗2F(z)) f (a∗2 b∗2 c) ≥ ∨ a∈F(x),c∈F(z) ( f (a) ∧ f (c) ) =     ∨ a∈F(x) f (a)  ∧   ∨ c∈F(z) f (c)     = F+ (f) (x) ∧ F+ (f) (z) . Hence ∧ γ∈(x∗1y∗1z) F+ (f) (γ) ≥ F+ (f) (x) ∧ F+ (f) (z) ∀x,y,z ∈ Q1. � Proposition 5.1. Let F be a strong set-valued homomorphism and let {fi}i∈I be a family of fuzzy bi- hyperideal of Q2. Then F − (∧ i∈I (fi) ) is a fuzzy bi-hyperideal of Q1. Int. J. Anal. Appl. 18 (6) (2020) 1010 Proof. Since every fi is a fuzzy bi-hyperideals for every i ∈ I, and for every x,y ∈ Q1, F− (∧ i∈I (fi) ) (α) = F− (∧ i∈I (fi) ) (x∨y) = (∧ i∈I F− (fi) ) (x∨y) = ∧ i∈I F− (fi) (x∨y) ≥ ∧ i∈I ( F− (fi) (x) ∧ F− (fi) (y) ) = {(∧ i∈I F− (fi) ) (x) ∧(∧ i∈I F− (fi) ) (y) } = F− (∧ i∈I fi ) (x) ∧ F− (∧ i∈I fi ) (y) . Hence ∧ α∈x∨y F− (∧ i∈I fi ) (α) ≥ F− (∧ i∈I fi ) (x) ∧ F− (∧ i∈I fi ) (y) ∀x,y ∈ Q1. Now, F− (∧ i∈I (fi) ) (β) = F− (∧ i∈I (fi) ) (x∗1 y) = (∧ i∈I F− (fi) ) (x∗1 y) = ∧ i∈I F− (fi) (x∗1 y) ≥ ∧ i∈I ( F− (fi) (x) ∧ F− (fi) (y) ) = {(∧ i∈I F− (fi) ) (x) ∧(∧ i∈I F− (fi) ) (y) } = F− (∧ i∈I fi ) (x) ∧ F− (∧ i∈I fi ) (y) . Hence ∧ β∈x∗1y F− (∧ i∈I fi ) (β) ≥ F− (∧ i∈I fi ) (x) ∧ F− (∧ i∈I fi ) (y) ∀x,y ∈ Q1. Int. J. Anal. Appl. 18 (6) (2020) 1011 Again since F is a strong set-valued homomorphism, and f is a fuzzy bi-hyperideal of Q2, so for any γ ∈ x∗1 y ∗1 z, we have, F− (∧ i∈I (fi) ) (γ) = ( F− ∧ i∈I (fi) ) (x∗1 y ∗1 z) = ∧ i∈I F− (fi) (x∗1 y ∗1 z) ≥ ∧ i∈I ( F− (fi) (x) ∧ F− (fi) (z) ) = {(∧ i∈I F− (fi) ) (x) ∧(∧ i∈I F− (fi) ) (z) } = F− (∧ i∈I fi ) (x) ∧ F− (∧ i∈I fi ) (z) . Hence ∧ γ∈x∗1y∗1z F− (∧ i∈I fi ) (γ) ≥ F− (∧ i∈I fi ) (x) ∧ F− (∧ i∈I fi ) (z) ∀x,y,z ∈ Q1. � For the following Theorem we define the set fα where α ∈ [0, 1] as following fα = {x ∈ Q | f (x) ≥ α} . Theorem 5.3. Let F be a strong set-valued homomorphism and f be a fuzzy bi-hyperideal of Q2. Then F− (f) (resp. F+ (f)) is a fuzzy bi-hyperideal of Q1 if and only if for each α ∈ [0, 1] , F− (fα) (resp. F+ (fα)), where fα 6= ∅, is a bi-hyperideal of Q1. Proof. Assume that F− (f) is a fuzzy bi-hyperideal of Q1. We need to show that F − (fα) is a bi-hyperideal of Q1. Let x1,x2 ∈ F− (fα) . Then F− (f) (x1) ≥ α and F− (f) (x2) ≥ α. But since F− (f) is a fuzzy bi-hyperideal, so ∧ z∈x1∨x2 F− (f) (z) ≥ F− (f) (x1) ∧ F− (f) (x2) ≥ α. Implies that F− (f) (z) ≥ α. Hence x1 ∨ x2 ⊆ F− (fα) . Let y ∈ F− (fα) , x ∈ Q1, and x ≤ y. Then F− (f) (x) ≥ F− (f) (y) ≥ α. Hence F− (f) (x) ≥ α. Hence x ∈ F− (fα) . Let y1,y2 ∈ F− (fα) , then F− (f) (y1) ≥ α and F− (f) (y2) ≥ α. Since F− (f) is a fuzzy bi-hyperideal of Q1, so we have ∧ z∈y1∗1y2 F− (f) (z) ≥ F− (f) (y1) ∧ F− (f) (y2) = α. Hence F− (f) (z) ≥ α, for all z ∈ y1 ∗1 y2, this implies that z ∈ F− (fα) . Hence y1 ∗1 y2 ⊆ F− (fα) . Now let u,v,w ∈ F− (fα) . Then F− (f) (u) ≥ α, F− (f) (v) ≥ α and F− (f) (w) ≥ α. Again since F− (f) is a fuzzy bi-hyperideal of Q1, so we have ∧ β∈u∗1v∗1w F− (f) (β) ≥ F− (f) (u) ∧ F− (f) (w) = α. Hence F− (f) (β) ≥ α. Thus u∗1 v ∗1 w ⊆ F− (fα) . Therefore F− (fα) is a bi-hyperideal of Q1. Conversely, assume that F− (fα) is a bi-hyperideal of Q1. We shall show that F − (f) is a fuzzy bi- hyperideal of Q1. For any x,y ∈ Q1, let α = F− (f) (x) ∧ F− (f) (y) ∈ range(F− (f)) . Then F− (f) (x) ≥ α and F− (f) (y) ≥ α. So x,y ∈ F− (fα) . Hence x∨y ⊆ F− (fα) . Consider Int. J. Anal. Appl. 18 (6) (2020) 1012 F− (f) (x∨y) = ∧ z∈F(x∨y) f (z) = ∧ z∈F(x)∨F(y) f (z) . Since z ∈ F (x) ∨F (y), there exist a ∈ F (x) and b ∈ F (y) such that z ∈ a∨ b. Hence F− (f) (x∨y) = ∧ a∨b∈F(x)∨F(y) f (a∨ b) ≥ ∧ a∈F(x),b∈F(y) ( f (a) ∧ f (b) ) =     ∧ a∈F(x) f (a)  ∧   ∧ b∈F(y) f (b)     = F− (f) (x) ∧ F− (f) (y) . Hence ∧ z∈x∨y F− (f) (z) ≥ F− (f) (x) ∧ F− (f) (y) ∀x,y ∈ Q1. Now for x,y ∈ F− (fα), we have x∗1 y ⊆ F− (fα) . Hence for β ∈ x∗1 y, we have F− (f) (β) ≥ α. Since, x,y ∈ F− (fα) , so F− (f) (x) ≥ α and F− (f) (y) ≥ α. Thus ∧ β∈x∗1y F− (f) (β) ≥ F− (f) (x) ∧ F− (f) (y) . Let x,y ∈ Q1 such that x ≤ y. If F− (f) (y) = 0 then F− (f) (x) ≥ F− (f) (y) . If F− (f) (y) = α then y ∈ F− (fα) . Since x ≤ y and F− (fα) is a bi-hyperideal of Q1, we have x ∈ F− (fα) . Then F− (f) (x) ≥ α = F− (f) (y) . Now let x,y,z ∈ F− (fα) . Then x∗1y∗1z ⊆ F− (fα) . Since x,z ∈ F− (fα) . Then F− (f) (x) ≥ α and F− (f) (z) ≥ α. So for any γ ∈ x∗1 y∗1 z, we have F− (f) (γ) ≥ α. Thus F− (f) (x) ∧ F− (f) (z) = α ≤∧ γ∈(x∗1y∗1z) F− (f) (γ). Therefore F− (f) is a fuzzy bi-hyperideal of Q1. � 6. Conclusion In the present paper, we introduced the notion of bi-hyperideals of hyperquantales. Furthermore we introduced the notions of fuzzy bi-hyperideals and generalized rough fuzzy bi-hyperideals of hyperquantales and their related properties is provided. 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